Riemann for Anti-Dummies: Part 29 : The Crimes of Klein

Riemann for Anti-Dummies Part 29


When working through the conceptions underlying Gauss’ 1799 proof of the fundamental theorem of algebra, or, Gauss’ discovery of the principles behind the division of the circle (to take only two examples), one is immediately confronted with the fact that these discoveries arise from explicitly anti-deductive methods of thinking. Most difficulties experienced by modern students attempting to work through these discoveries, are rooted in the tendency of those individuals to revert to ingrained habits of deductive thinking, just at the point when only an explicitly anti-deductive, creative leap will do. “Where’s the cube in Archytus’ construction?”; “What exactly is Gauss trying to prove?”; “I understand what you said, but I still don’t understand what it means,” are some common symptoms of this affliction.

The serious person can take heart that such symptoms need not indicate an incurable condition, but it is only the recurring effects of the malicious teaching methods most people today have suffered through. It may be helpful to those suffering from these effects, to take a clinical look at how this “deductivizing” was introduced into modern educational practices by G.W.F. Hegel’s grandson-in-law, Felix Klein. As a talented mathematician, Klein was not as radical a reductionist or as openly fascistic as Russell, Kronecker or Helmholtz. Yet his method was pure Bogomilism, nevertheless. Rather than try and obliterate the creative discoveries of Leibniz, Gauss and Riemann, Klein adopted a seemingly “middle ground” so to speak, in which the discoveries were stripped of their creative insight, and re-cast in deductive, i.e. impotent, form.

While Klein had an extensive influence over the teaching methods of a wide domain of scientific subjects, it is sufficient, for our purposes at this moment, to look at his treatment of Gauss’ early discoveries, to obtain the clinical benefit of freeing those individuals, who, knowingly or not, have been victimized by Klein’s crime.

As discussed in the recent pedagogicals of this series, Gauss’ early discoveries have their origin in the paradoxes arising from the investigations of “powers” as that concept is defined by Plato, and how these paradoxes arise in the classical problems of doubling the cube and trisecting an angle. For Plato, Cusa, Kepler, Leibniz, Kaestner, Gauss, Riemann, these investigations led into the deepest questions concerning the relationship of man to the universe. However, in his 1895 lectures, “Famous Problems of Elementary Geometry”, Klein reduces these problems to the following, which will seem uneasily familiar to most students today:

“In all these problems the ancients sought in vain for a solution with straight edge and compasses, and the celebrity of these problems is due chiefly to the fact that their solution seemed to demand the use of appliances of a higher order…”

This already is complete fraud. Plato’s circle did not consider the straight edge and compass as “appliances”, but as Kepler summarizes the question in the first book of the “Harmonies of the World”, the question under investigation was the “knowability” of magnitudes. That is, which magnitudes were “knowable” from the circumference and diameter of a circle, and which were “unknowable”.

Klein continues, “At the outset we must insist upon the difference between practical and theoretical constructions. For example, if we need a divided circle as a measuring instrument, we construct it simply on trial. Theoretically, in earlier times, it was possible (i.e. by the use of straight edge and compasses) only to divide the circle into a number of parts represented by 2n, 3 and 5 and their products. Gauss added other cases by showing the possibilty of the division into parts where p is a prime number of the form p = (22p) + 1, and the impossibility for all other numbers. No practical advantage is derived from these results; the significance of Gauss’ developments is purely theoretical.”

Klein’s separation of the theoretical and practical is pure evil Bogomilism, in addition to being a fraud. One need look no further, than Erathosthenes’ account of the history of the duplication of the cube, as reported by Theon of Smyrna:

“Eratosthenes in his work entitled “Plotinicus” relates that, when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an altar double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.”

Where is the separation of the theoretical from the practical in Eratosthenes account? Was it purely a theoretical matter, that the Delians had become so morally corrupt by their neglect of the cognitive powers of the mind, that they had become victims of a deadly plague?

As the Thirty Years War began to unfold in full horror, Kepler, on the occasion of the twenty-fifth anniversary of the publication of his “Mysterium Cosmographicum”, invoked the “practical” benefits of the power of cognition, “would that even now indeed there may still, after the reversal of Austrian affairs which followed, be a place for Plato’s oracular saying. For when Greece was on fire on all sides with a long civil war, and was troubled with all the evils which usually accompany civil war, he was consulted about a Delian Riddle, and was seeking a pretext for suggesting salutary advice to the peoples. At length he replied that, according to Apollo’s opinion Greece would be peaceful if the Greeks turned to geometry and other philosophical studies, as these studies would lead their spirits from ambition and other forms of greed, out of which wars and other evils arise, to the love of peace and to moderation in all things.”

And Gauss, himself, when installed as head of the Goettingen University Observatory, pronounced that the political troubles that had befallen Europe at that time, arose from a contempt for purely cognitive discoveries.

Klein is deadly wrong. Gauss’ discoveries were not purely theoretical. Recognizing that is crucial to being able to grasp elementary mathematics from a truly advanced, (LaRouchian) standpoint.